Discussion Overview
The discussion revolves around the mathematical treatment of an electron in a metal subjected to a time-dependent force, specifically focusing on the contributions to momentum change over a small time interval. Participants explore the implications of including higher-order terms in the momentum equation as presented in Ashcroft and Mermin's text.
Discussion Character
- Technical explanation
- Mathematical reasoning
Main Points Raised
- One participant questions the origin of the dt^2 term in the momentum change equation, suggesting that the change in momentum can be simplified to f(t)dt based on Newton's second law.
- Another participant explains that the dt^2 term arises because the force f(t) is not constant over the interval, indicating a relationship to the derivative of f(t).
- A later reply reiterates the point about the non-constancy of f(t) and questions the necessity of the dt^2 term, suggesting that Newton's second law should suffice for determining momentum change.
- One participant concludes that using a first-order Taylor series expansion for the force at t+dt reveals how the additional dt term emerges in the calculations.
Areas of Agreement / Disagreement
Participants express differing views on the necessity and origin of the dt^2 term, with some advocating for its inclusion based on the non-constancy of the force, while others maintain that Newton's second law alone should suffice for the analysis. The discussion remains unresolved regarding the best approach to take.
Contextual Notes
The discussion highlights the dependence on the assumptions made about the force f(t) and the mathematical treatment of momentum change, particularly in relation to Taylor series approximations and the limits of Newton's second law.